diff --git a/src/linalg/cholesky.rs b/src/linalg/cholesky.rs
index c4049504..d0a9918c 100644
--- a/src/linalg/cholesky.rs
+++ b/src/linalg/cholesky.rs
@@ -147,7 +147,7 @@ where
     }
 
     /// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `v`,
-    /// performs a rank one update such that we end up with the decomposition of `M + sigma * v*v^*`.
+    /// performs a rank one update such that we end up with the decomposition of `M + sigma * v*v.adjoint()`.
     pub fn rank_one_update<R2: Dim, S2>(&mut self, x: &Matrix<N, R2, U1, S2>, sigma: N::RealField)
     where
         S2: Storage<N, R2, U1>,
@@ -156,27 +156,31 @@ where
     {
         // for a description of the operation, see https://en.wikipedia.org/wiki/Cholesky_decomposition#Updating_the_decomposition
         // heavily inspired by Eigen's implementation https://eigen.tuxfamily.org/dox/LLT_8h_source.html
-        // TODO use unsafe { *matrix.get_unchecked((j, j)) }
         let n = x.nrows();
-        let mut temp = x.clone_owned();
+        let mut x = x.clone_owned();
         let mut beta = crate::one::<N::RealField>();
         for j in 0..n {
-            let ljj = N::real(self.chol[(j, j)]);
-            let dj = ljj * ljj;
-            let wj = temp[j];
-            let swj2 = sigma * N::modulus_squared(wj);
-            let gamma = dj * beta + swj2;
-            let nljj = (dj + swj2 / beta).sqrt();
-            self.chol[(j, j)] = N::from_real(nljj);
-            beta += swj2 / dj;
+            let diag = N::real(unsafe { *self.chol.get_unchecked((j, j)) });
+            let diag2 = diag * diag;
+            let xj = unsafe { *x.get_unchecked(j) };
+            let sigma_xj2 = sigma * N::modulus_squared(xj);
+            let gamma = diag2 * beta + sigma_xj2;
+            let new_diag = (diag2 + sigma_xj2 / beta).sqrt();
+            unsafe { *self.chol.get_unchecked_mut((j, j)) = N::from_real(new_diag) };
+            beta += sigma_xj2 / diag2;
             // Update the terms of L
             if j < n {
-                for k in (j + 1)..n {
-                    temp[k] -= (wj / N::from_real(ljj)) * self.chol[(k, j)];
-                    if gamma != crate::zero::<N::RealField>() {
-                        self.chol[(k, j)] = N::from_real(nljj / ljj) * self.chol[(k, j)]
-                            + (N::from_real(nljj * sigma / gamma) * N::conjugate(wj)) * temp[k];
-                    }
+                let mut xjplus = x.rows_range_mut(j + 1..);
+                let mut col_j = self.chol.slice_range_mut(j + 1.., j);
+                // temp_jplus -= (wj / N::from_real(diag)) * col_j;
+                xjplus.axpy(-xj / N::from_real(diag), &col_j, N::one());
+                if gamma != crate::zero::<N::RealField>() {
+                    // col_j = N::from_real(nljj / diag) * col_j  + (N::from_real(nljj * sigma / gamma) * N::conjugate(wj)) * temp_jplus;
+                    col_j.axpy(
+                        N::from_real(new_diag * sigma / gamma) * N::conjugate(xj),
+                        &xjplus,
+                        N::from_real(new_diag / diag),
+                    );
                 }
             }
         }
diff --git a/tests/linalg/cholesky.rs b/tests/linalg/cholesky.rs
index b04ed402..ea8402a3 100644
--- a/tests/linalg/cholesky.rs
+++ b/tests/linalg/cholesky.rs
@@ -82,13 +82,13 @@ macro_rules! gen_tests(
                     let mut m = RandomSDP::new(U4, || random::<$scalar>().0).unwrap();
                     let x = Vector4::<$scalar>::new_random().map(|e| e.0);
 
-                    // TODO this is dirty but $scalar appears to not be a scalar type in this file
+                    // this is dirty but $scalar is not a scalar type (its a Rand) in this file
                     let zero = random::<$scalar>().0 * 0.;
                     let one = zero + 1.;
                     let sigma = random::<f64>(); // needs to be a real
                     let sigma_scalar = zero + sigma;
 
-                    // updates cholesky decomposition and reconstructs m
+                    // updates cholesky decomposition and reconstructs m updated
                     let mut chol = m.clone().cholesky().unwrap();
                     chol.rank_one_update(&x, sigma);
                     let m_chol_updated = chol.l() * chol.l().adjoint();